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About Ratio & Proportion — Class 10 ICSE

Apply componendo-dividendo, mean proportion, and continued proportion to solve problems. This topic is part of the ICSE Class 10 mathematics syllabus (chapter: Chapter 15). On this page you can practice 58 questions across three difficulty levels — 20 easy, 19 medium, and 19 hard — each with a visual step-by-step solution, plus a timed 34-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

What you'll learn in Ratio & Proportion

  • Unit 1: Introduction to Ratios – Comparing Quantities
  • Unit 2: Understanding Proportion – Equality of Ratios
  • Unit 3: Componendo and Dividendo – Powerful Transformations
  • Unit 4: Mean and Continued Proportion – Special Cases
  • Unit 5: Advanced Problem Solving & Lesson Summary

Interactive lesson · about 15 minutes · checkpoint question after every unit

Ratio & Proportion — solved examples for Class 10 ICSE

Example 1easy

Which of the following statements correctly describes a ratio?
  1. A)A ratio always compares two quantities using addition.
  2. B)A ratio is a comparison of two quantities of different kinds.
  3. C)A ratio is a comparison of two quantities of the same kind by division.
  4. D)A ratio compares more than two quantities simultaneously.

Step-by-step solution

  1. A ratio expresses how many times one quantity contains another of the same kind, or what part one quantity is of another. It's fundamentally a comparison by division.
  2. Option A is incorrect as ratios use division. Option B is incorrect because quantities must be of the same kind (or convertible to the same units). Option D is incorrect as a basic ratio compares two quantities, though compound ratios involve more.

Answer: A ratio is a comparison of two quantities of the same kind by division.

Example 2medium

What is the mean proportional between (a - b)² and (a + b)²?
  1. A)a² - b²
  2. B)a² + b²
  3. C)(a - b)(a + b)
  4. D)2ab

Step-by-step solution

  1. Let the two numbers be x = (a - b)² and y = (a + b)².
  2. The mean proportional (m) is given by m = √(x × y).
  3. m = √[ (a - b)² × (a + b)² ]
  4. m = √[ ((a - b)(a + b))² ] = (a - b)(a + b) = a² - b².

Answer: a² - b²

Example 3hard

If (x³+y³)/(x³-y³) = (a³+b³)/(a³-b³) and x, y, a, b are positive, then x/y is equal to:
  1. A)a/b
  2. B)b/a
  3. C)a³/b³
  4. D)√(a/b)

Step-by-step solution

  1. Given (x³+y³)/(x³-y³) = (a³+b³)/(a³-b³).
  2. Applying Componendo and Dividendo: ((x³+y³)+(x³-y³))/((x³+y³)-(x³-y³)) = ((a³+b³)+(a³-b³))/((a³+b³)-(a³-b³)).
  3. This simplifies to (2x³)/(2y³) = (2a³)/(2b³), which means x³/y³ = a³/b³.
  4. Taking the cube root of both sides, we get x/y = a/b.

Answer: a/b

Practice questions on Ratio & Proportion

  1. Q1.easy

    If a, b, c, d are in proportion, which of the following statements is always true?
    1. A)a : b = c : d
    2. B)a + b = c + d
    3. C)a × c = b × d
    4. D)a / c = b / d
    Show answer

    Answer: a : b = c : d

    Hint: Remember the fundamental definition of four quantities being in proportion.

  2. Q2.easy

    Find the mean proportion between 9 and 16.
    1. A)10
    2. B)12
    3. C)12.5
    4. D)14
    Show answer

    Answer: 12

    Hint: For two numbers 'a' and 'b', the mean proportion 'x' satisfies the condition a : x = x : b.

  3. Q3.easy

    What is the third proportion to 4 and 10?
    1. A)16
    2. B)20
    3. C)24
    4. D)25
    Show answer

    Answer: 25

    Hint: If 'c' is the third proportion to 'a' and 'b', then a : b = b : c.

  4. Q4.medium

    If x, 18, 54 are in continued proportion, find the value of x.
    1. A)3
    2. B)6
    3. C)9
    4. D)12
    Show answer

    Answer: 6

    Hint: In continued proportion, if a, b, c are in continued proportion, then a/b = b/c, which implies b² = ac.

  5. Q5.medium

    Find the fourth proportional to (a² - b²), (a + b), and (a - b).
    1. A)1
    2. B)a + b
    3. C)a - b
    4. D)(a - b)²
    Show answer

    Answer: 1

    Hint: If a, b, c, d are in proportion, then a/b = c/d. Set up the proportion and solve for the unknown fourth term.

  6. Q6.medium

    If (√(x+1) + √(x-1)) / (√(x+1) - √(x-1)) = 3, what is the value of x?
    1. A)5/3
    2. B)3/5
    3. C)1/3
    4. D)3
    Show answer

    Answer: 5/3

    Hint: Apply the Componendo and Dividendo rule to the given equation. This will help simplify the square root terms.

  7. Q7.hard

    If (x+y)/z = (y+z)/x = (z+x)/y = k and x+y+z ≠ 0, then the value of k is:
    1. A)1
    2. B)2
    3. C)3
    4. D)-1
    Show answer

    Answer: 2

    Hint: Remember the Addendo property for equal ratios: if multiple ratios are equal, then each ratio is also equal to the sum of all numerators divided by the sum of all denominators.

  8. Q8.hard

    Three numbers a, b, c are in continued proportion. If a+b = 10 and b+c = 30, find the value of c-a.
    1. A)10
    2. B)15
    3. C)20
    4. D)25
    Show answer

    Answer: 20

    Hint: For numbers in continued proportion, the square of the middle term is equal to the product of the first and third terms (b²=ac). Use the given sums to express 'a' and 'c' in terms of 'b'.

  9. Q9.hard

    Find the mean proportion between (x² - 2xy + y²)/(x + y) and (x³ + y³)/(x² - xy + y²). (Assume x ≠ y, x+y ≠ 0)
    1. A)x-y
    2. B)x+y
    3. C)x²-y²
    4. D)1/(x-y)
    Show answer

    Answer: x-y

    Hint: The mean proportion 'm' between two quantities A and B is √(A × B). Simplify each given algebraic expression using identities before multiplying.

These are 9 of the 58 questions available for Ratio & Proportion. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.